Theorem splcl 12390

Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)

Assertion

Ref	Expression
splcl	|- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) e. Word A )

Proof of Theorem splcl

Dummy variables s b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2979	. . . 4 |- ( S e. Word A -> S e. _V )
2		otex 4554	. . . 4 |- <. F , T , R >. e. _V
3		id 22	. . . . . . . 8 |- ( s = S -> s = S )
4		fveq2 5688	. . . . . . . . . 10 |- ( b = <. F , T , R >. -> ( 1st ` b ) = ( 1st ` <. F , T , R >. ) )
5	4	fveq2d 5692	. . . . . . . . 9 |- ( b = <. F , T , R >. -> ( 1st ` ( 1st ` b ) ) = ( 1st ` ( 1st ` <. F , T , R >. ) ) )
6	5	opeq2d 4063	. . . . . . . 8 |- ( b = <. F , T , R >. -> <. 0 , ( 1st ` ( 1st ` b ) ) >. = <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. )
7	3, 6	oveqan12d 6109	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) = ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) )
8		simpr 458	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> b = <. F , T , R >. )
9	8	fveq2d 5692	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 2nd ` b ) = ( 2nd ` <. F , T , R >. ) )
10	7, 9	oveq12d 6108	. . . . . 6 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) = ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) )
11		simpl 454	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> s = S )
12	8	fveq2d 5692	. . . . . . . . 9 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 1st ` b ) = ( 1st ` <. F , T , R >. ) )
13	12	fveq2d 5692	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 2nd ` ( 1st ` b ) ) = ( 2nd ` ( 1st ` <. F , T , R >. ) ) )
14	11	fveq2d 5692	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( # ` s ) = ( # ` S ) )
15	13, 14	opeq12d 4064	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. = <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. )
16	11, 15	oveq12d 6108	. . . . . 6 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) = ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) )
17	10, 16	oveq12d 6108	. . . . 5 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) concat ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
18		df-splice 12230	. . . . 5 |- splice = ( s e. _V , b e. _V |-> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) concat ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) )
19		ovex 6115	. . . . 5 |- ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. _V
20	17, 18, 19	ovmpt2a 6220	. . . 4 |- ( ( S e. _V /\ <. F , T , R >. e. _V ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
21	1, 2, 20	sylancl 657	. . 3 |- ( S e. Word A -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
22	21	adantr 462	. 2 |- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
23		swrdcl 12311	. . . . 5 |- ( S e. Word A -> ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A )
24	23	adantr 462	. . . 4 |- ( ( S e. Word A /\ R e. Word A ) -> ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A )
25		ot3rdg 6592	. . . . . 6 |- ( R e. Word A -> ( 2nd ` <. F , T , R >. ) = R )
26	25	adantl 463	. . . . 5 |- ( ( S e. Word A /\ R e. Word A ) -> ( 2nd ` <. F , T , R >. ) = R )
27		simpr 458	. . . . 5 |- ( ( S e. Word A /\ R e. Word A ) -> R e. Word A )
28	26, 27	eqeltrd 2515	. . . 4 |- ( ( S e. Word A /\ R e. Word A ) -> ( 2nd ` <. F , T , R >. ) e. Word A )
29		ccatcl 12270	. . . 4 |- ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A /\ ( 2nd ` <. F , T , R >. ) e. Word A ) -> ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A )
30	24, 28, 29	syl2anc 656	. . 3 |- ( ( S e. Word A /\ R e. Word A ) -> ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A )
31		swrdcl 12311	. . . 4 |- ( S e. Word A -> ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A )
32	31	adantr 462	. . 3 |- ( ( S e. Word A /\ R e. Word A ) -> ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A )
33		ccatcl 12270	. . 3 |- ( ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A /\ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A ) -> ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. Word A )
34	30, 32, 33	syl2anc 656	. 2 |- ( ( S e. Word A /\ R e. Word A ) -> ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. Word A )
35	22, 34	eqeltrd 2515	1 |- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) e. Word A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1364   e. wcel 1761   _Vcvv 2970   <.cop 3880   <.cotp 3882   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   0cc0 9278   #chash 12099  Word cword 12217   concat cconcat 12219   substr csubstr 12221   splice csplice 12222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-concat 12227  df-substr 12229  df-splice 12230

This theorem is referenced by:  psgnunilem2  15994  efglem  16206  efgtf  16212  frgpuplem  16262